−
Lecture 21 :
Finding bases:
Expanding a linearly independent
set to a basis of a vector space.
(Refers to section 4.4)
Expectations:
1.
Apply an algorithm to
expand
a linearly independent set of vectors to a basis of
V
.
21.1
Proposition
−
Let
U
=
{
v
1
,
v
2
,
....,
v
k
} be
a linearly independent set of vectors in the
vector space
V
such that span(
v
1
,
v
2
,
....,
v
k
) is NOT all of
V
. Then, for any vector
v
in
V
/span(
U
),
the set {
v
1
,
v
2
,
....,
v
k
,
v
} forms a linearly independent set.
Proof (Outline):
o
Let
U
= {
v
1
,
v
2
,
....,
v
k
} be linearly independent and
v
be a vector not in the span of
U
.
o
Suppose
α
1
v
1
+
α
2
v
2
+..., +
α
k
v
k
+
α
v
=
0
.
o
We claim
α
= 0:
.
±
Suppose
α ≠
0.
±
Then
v
=
−
(
α
1
/
α
)
v
1
+
−
(
α
2
/
α
)
v
2
+
, .
..,
+
−
(
α
k
/
α
)
v
k
∈
span(
U
), a
contradiction.
±
Thus
α
= 0.
o
Also
α
1
v
1
+
α
2
v
2
+..., +
α
k
v
k
=
0
implies
α
i
= 0 for all
i.
Why?
o
Thus
α
i
= 0 for all
i
and
α
= 0
.
o
Thus {
v
1
,
v
2
,
....,
v
k
,
v
} forms a linearly independent set. &
21.2.Consequence of Proposition 21.1
−
For any finite dimensional vector space
V
,
a
linearly independent set of vectors can be completed to a basis for V